Analytical Study of Damped Standing Longitudinal MHD Waves in Flowing Coronal Loops: The Effect of Thermal Conduction

The longitudinal magnetohydrodynamic (MHD) oscillations of coronal loops have been investigated in dissipative flowing loops. Thermal conduction has been considered as the damping mechanism of the wave. We aim to construct the damped longitudinal waves by superposing two propagating waves that propagate in opposite directions. The two propagating components must have the same oscillation frequencies and damping rates, which has been described impossible by some authors, but we have used a technique to overcome this difficulty. The equations of motion are combined to obtain a differential equation for the velocity perturbation. Using the weak damping condition, the perturbation method is used to solve the dispersion relation. In the leading order approximation, the oscillation frequency of the standing waves is determined. In the first-order approximation, we let both the oscillation frequency and wavelength of the propagating waves be perturbed due to the presence of thermal conduction, which enables us to determine the damping rate of the standing waves. Our results show that the plasma flow is an essential parameter in determining the effectiveness of the damping mechanism. Also, the exact solutions of the dispersion relation have been determined without using weak damping assumption. Interestingly the two solutions are the same. Introducing plasma flow to the coronal loop causes the period ratio of the fundamental mode to the first overtone to decrease more.


Introduction
It is known that the solar atmosphere is dynamic in nature and magnetically structured (Vaiana et al. 1973;Schrijver et al. 1999).The small-scale structure of the solar corona consists of a variety of distinct magnetic features, for example, open flux tubes, coronal loops, prominences, etc.These onedimensional structures play a pronounced role in coronal dynamics since they support various types of MHD oscillations and waves that are believed to have contribution to the solution of the problem of solar coronal heating (see the reviews Erdelyi 2008 andErdélyi &Taroyan 2008).Observations show that various modes of oscillations are present everywhere in the solar atmosphere (De Pontieu et al. 2007;Van Doorsselaere et al. 2008 andJess et al. 2009).Many of the mentioned reports indicate that the observed oscillations or waves are seen to be damped strongly.The commonly suggested and effective mechanism for wave damping is resonant absorption, which has been studied by Ruderman & Roberts (2002) and Goossens et al. (2002).This mechanism is used to explain the damping of the kink waves.Other effective damping mechanisms are compressive viscosity and thermal conduction, which has been investigated by Al-Ghafri et al. (2014) and Bahari & Shahhosaini (2018) and has been used to study the damping of the longitudinal waves.Ruderman (2011), Bahari (2017a), andAl-Ghafri et al. (2014) argued that the cooling of the background plasma causes the wave amplitude to amplify.
Resonant absorption of propagating MHD waves has been studied by some authors, e.g., Goossens et al. (1992), Ruderman & Roberts (2002), Goossens et al. (2002), Shukhobodskiy & Ruderman (2018), Bahari (2018), and Bahari (2020).One of the important features of flux tubes in the solar atmosphere is the presence of plasma flow in flux tubes.The flow has been observed in active regions by Brekke et al. (1997), Winebarger et al. (2002), Doyle et al. (2006), Tian et al. (2009), Shibata et al. (2007), and Terradas et al. (2008).In the case of the presence of the plasma flow or magnetic twist, which break down the symmetry of the loop between the waves that propagate in the two opposite directions, the damping of standing waves is not straightforward.This is because, as stated by Terradas et al. (2010), Pandey et al. (2012), and Ruderman & Petrukhin (2019), if we consider the standing wave as the superposition of two propagating waves, then it is impossible to determine the same oscillation frequencies and damping rates for both propagating waves.Because of this difficulty, many works have been done regarding the standing waves in the presence of plasma flow and magnetic twist but these works have refused the investigation the damping of the waves, e.g., Ruderman (2007Ruderman ( , 2010)), Terradas et al. (2011), Bahari (2017b), and Bahari & Jahan (2020).Also all of the authors who studied damped waves have studied only the damping of the propagating waves, except Ruderman & Petrukhin (2019) who investigated the damping of standing MHD kink waves in flowing loops, but their analytical treatment for studying the damping of the kink waves involves much mathematics.
The observation of slow MHD waves has been reported by Wang et al. (2005) and Krishna Prasad et al. (2014).The propagation and damping of slow MHD waves has been studied by Nakariakov et al. (2000), De Moortel & Hood (2003), and Verwichte et al. (2008).Doppler velocity oscillations induced from flares for the first time detected in hot loops by Solar and Heliospheric Observatory/Solar Ultraviolet Measurement of Emitted Radiation (SUMER) spectrometer (called SUMER oscillations) and reported by Wang et al. (2002) and Wang et al. (2003).These oscillations have been interpreted by Ofman & Wang (2002) as the standing slow waves.Also Kumar et al. (2013) and Kumar et al. (2015) used Solar Dynamics Observatory/Atmospheric Imaging Assembly data and reported longitudinal standing waves in flaring flux tubes with oscillation properties approximately the same as those of SUMER oscillations.The slow-mode magnetoacoustic waves have been reviewed by Wang et al. (2021).Regarding the longitudinal waves, it has been argued that the main dissipative mechanism is thermal conduction.However, it has been shown by Sigalotti et al. (2007) that thermal conduction cannot be responsible for the rapid observed damping.Bahari & Shahhosaini (2018) included the effect of compressive viscosity and concluded that in hot loops the compressive viscosity can be as effective as thermal conduction in damping longitudinal waves.
In the current work, we examine the effect of plasma flow on the oscillation frequency and damping rate of standing longitudinal waves in coronal loops.We use a novel method to determine the oscillation frequency and damping rate of standing waves.Our aim is as follows: 1. Contrary to the conclusions made earlier, we want to consider the damped standing waves as the superposition of two propagating waves that propagate in opposite directions.2. We investigate for the first time the effect of the plasma flow on the oscillation properties of standing longitudinal waves.
The paper is organized as follows: in the next section the tube model and equations of motion are presented, in Section 3 the dispersion relation is determined and is solved in the leading order approximation, in Section 4 the solution of the dispersion relation in the first-order approximation and the oscillation properties of the waves are presented, and Section 5 is devoted to conclusions.

The Model of the Tube and Equations of Motion
We study standing longitudinal MHD waves in flowing hot coronal flux tubes.To do this, the coronal loop is assumed to be a cylindrical tube with constant cross section and purely axial magnetic field.The flux tube is assumed not to be stratified and the variation of the equilibrium plasma density ρ 0 is negligible along the loop.This assumption is relevant in hot or short coronal flux tubes in which the density scale height is larger than the height of the loop apex.It is assumed that the longitudinal oscillations of the plasma do not cause any perturbation in the magnetic field, so the cross section of the loop does not change during the oscillations.The cross-section diameter of the tube is assumed to be much smaller than the curvature radius of the loop; hence the effect of the loop curvature is neglected.The background temperature and pressure of the loop are assumed to be time-independent and homogeneous along the loop and are denoted by T 0 and P 0 respectively.The loop is studied using a cylindrical coordinate system with the z-axis to coincide with the loop axis and the footpoints of the loop are denoted by z = 0 and z = L.
It is assumed that the stationary plasma flow with constant flow speed U enters the loop from one of the footpoints and exits from the other footpoint.For propagating waves the oscillation frequency and damping rate of the wave in flowing flux tubes depend on the direction of the flow; in other words, the oscillation frequency and damping rate of forward and backward waves are affected by plasma flow in different manners.By the forward (backward) waves we mean the waves that propagate in the direction (opposite direction) of the plasma flow.In the current work, we focus on the standing waves that are the superposition of the forward and backward waves; hence, as we see in Sections 3 and 4, in the current case the oscillation frequency and damping rate of of the waves do not depend on the direction of the plasma flow.Thermal conductivity is considered as the damping mechanism of the waves.The linearized equations of motion of the longitudinal perturbations of the loop are given as r r z shows the total differentiation, ρ 1 , v, T 1 , and P 1 represent the perturbations of plasma density, plasma velocity, temperature, and thermal pressure respectively, κ ∥ = κ 0 T 5/2 Wm −1 deg −1 is the thermal conduction along the magnetic field lines, and m ˜is the mean molecular weight.Note that, as stated by Spitzer (1962), the thermal conduction is negligible across the magnetic field lines; hence the thermal conduction is assumed to act in the direction of the zaxis; also κ 0 = 10 −11 in mks units (De Moortel & Hood 2003).The adiabatic sound speed is defined by g r = c P s 0 0 in which γ is the ratio of the specific heats.Equations (1)-( 4) can be combined to give a partial differential equation for the velocity perturbation Because the footpoints of the loop are laid in the dense photosphere, the appropriate boundary conditions of the wave are In the next two sections Equation (5) is simplified for standing longitudinal waves in the coronal loop and its eigenvalues and eigenfunctions are determined using the perturbation method.

Dispersion Relation: Leading Order Approximation
Because the background quantities of the loop are independent of time and the z-coordinate, it is possible to Fourier analyze the velocity perturbation of the propagating waves as v (z, t) ∝ e i(kz−ωt) , in which k and ω are the wavenumber and oscillation frequency respectively.After substituting this expression for v(z.t) in Equation (5), we obtain an equation for k, ω and the parameters of the loop, i.e., the dispersion relation for the propagating waves: The dispersion relation can be written in the following form: Here α and β are defined in terms of thermal conduction as In static tubes, the standing wave can be considered as the superposition of two propagating waves that have the same oscillation frequencies and wavenumbers and propagate in opposite directions.In this case, the flux tube has directional symmetry meaning that regarding the propagation of the longitudinal waves, the two directions of the tube are the same; we say that in this case, the loop has directional symmetry.In the flowing coronal loop the propagating waves that propagate in the direction of the plasma flow and in the opposite direction of the plasma flow are called forward and backward waves respectively.Since in our model the plasma flows in the loop, it breaks down the directional symmetry of the loop, and we expect that the forward and backward waves with the same oscillation frequencies no longer have the same wavenumbers too.In this case, to determine the oscillation frequency of the wave, the leading order approximation we use the method suggested by Bahari (2017b), Ruderman & Petrukhin (2019), and Bahari & Jahan (2020).In these works, in the absence of directional symmetry, the standing wave is considered as the superposition of two waves with the same oscillation frequencies but different wavenumbers.According to the fact that thermal conductivity is small in coronal loops, in comparison to undamped waves, it introduces small corrections to the eigenfunction v(z) and ω as the corresponding eigenvalue.Hence we can solve Equation (5) using the perturbation method.To use the perturbation method, we write the quantities k, ω, and v(z, t) as the sum of the zeroth and firstorder terms Our results show that the leading order values of k and ω are real while the first-order corrections to these quantities are imaginary; hence we denote the leading order and first-order of these quantities by the subscripts r and i respectively.Also, the superscripts (0) and (1) show the leading order and first-order terms of the eigenfunction respectively.In the leading order approximation, i.e., for κ ∥ = 0 we obtain undamped oscillations, whose frequencies can be determined from the leading order approximation of Equation ( 8) This equation has two physically acceptable solutions for k r w w Here k r+ and k r− are the longitudinal wavenumbers of the waves that propagate in the positive and negative z-directions respectively.These solutions are acceptable in the sense that they reduce to the wavenumbers of the propagating waves that constitute the standing wave in the case of the static loop.
Hence the eigenfunction of longitudinal oscillations in the leading order approximation becomes In this equation n = 1, 2, L is a positive integer that indicates the longitudinal mode number of the wave.The case n = 1 corresponds to the axial fundamental mode, n = 2 to the first overtone, and so on.It is a common procedure to consider a standing MHD wave as the superposition of two propagating waves.For propagating waves the wavenumbers k r± are continuous variables, while for standing waves the values of k r ± are subject to the footpoint condition given by Equation (15), which results in the axial mode number n of the standing waves.This method has been used by Li et al. (2013).In the absence of the plasma flow, the oscillation frequency of the fundamental mode given by Equation ( 16) 2 s , which is as expected because the wave period is the time required for the wave to travel between the two footpoints twice.In the case of flowing tubes, the oscillation frequency decreases, which can be understood simply.In this case too, the period time is the time required for the wave to travel between the footpoints of the loop twice.The Doppler shifted phase speed of the wave is c s + U and c s − U for forward and backward waves, which gives the period time equal to . This is in agreement with Equation (16).

First-order Perturbation: Damped Waves
To our knowledge, this is for the first time that the damping rate of the standing longitudinal waves in coronal loops in the absence of the directional symmetry has been studied.Mathematically this problem is similar to the problem of determining the damping rate of the standing kink waves in flowing or twisted loops.As stated by Terradas et al. (2010) and Pandey et al. (2012), in the presence of plasma flow (or in the presence of magnetic twist in the case of various MHD waves) the directional symmetry is broken and the damping rate of the forward and backward waves with the same oscillation frequencies, which construct a standing wave, can no longer be equal to each other.It is stated by these authors that in this case the standing wave cannot be considered as the superposition of the two propagating waves.To overcome this problem, i.e., in order to consider the damped standing wave as the superposition of two propagating waves with the same oscillation frequencies and damping rates, we let the two propagating waves that have the same oscillation frequencies to have wavenumbers slightly different from k r± , under the restriction that they must still construct a standing wave and satisfy the boundary conditions.In other words we assume that the damping mechanism in addition to the oscillation frequency causes perturbations in the wavenumbers too.This permits us to obtain the same oscillation frequencies and the same damping rates for the two propagating waves.
In order to determine the damping rate of the waves, one may use a mathematical method like that used by Al-Ghafri et al. (2014), for the appropriate case of constant temperature and flowing loop.If the mentioned method is used here it would include three steps: first, writing the differential equation governing the waves, i.e., Equation (5) in the leading order approximation and solving it, the result of which would be the same as that presented in Section 3. The second step is to write Equation (5) in the first-order approximation.In the third step the compatibility condition of the first-order differential equation is used to determine the damping rate of the wave without obtaining the first-order eigenfunction v (1) (z).In the present work, a method like that of Al-Ghafri et al. (2014) fails to determine the damping rate, because using the mentioned method here would include the calculation of some integrals for which the first-order eigenfunction v (1) (z) must be known; however, it is unknown at that stage.Instead, we focus on the exact dispersion relation that is given by Equation (8) and write it in the first-order approximation, which gives The perturbation to the wavenumbers of the propagating waves can be determined as In this equation, k i+ and k i− are the first-order corrections to the wavenumber of the waves propagating in the positive and negative z-directions respectively.Hence the eigenfunction of the longitudinal waves up to the first-order perturbation is Imposing the boundary condition Equation ( 6) at z = 0 to the solution given by Equation (20), similar to the leading order approximation one obtains = - and inserting the velocity perturbation Equation (20) in the boundary condition at z = L, given by Equation (6) gives Substituting k i+ and k i− from Equation (18) into Equation (22) after some simplification gives the damping rate of the standing longitudinal waves as In the limit of vanishing flow speed, this result reduces to the corresponding results determined by Al-Ghafri et al. (2014) and Bahari & Shahhosaini (2018).
In Figure 1 the ratio of the damping time τ = 1/|ω i | to the period time P = 2π/ω r has been plotted as a function of the flow speed.In this figure, the background plasma temperature is T 0 = 10 6 K for the solid curve, and T 0 = 1.5 × 10 6 K for the dashed curve and κ 0 = 10 (−11) .The sound speed c s is considered as the unit of the flow speed in all the figures.The quantities that are plotted as functions of flow speed in all the figures are dimensionless, so it is not necessary to choose a specific unit for the oscillation frequency or time.It is clear from Figure 1 that for T = 10 6 K in the absence of plasma flow the damping time is about 13.6 times the period time, and as the flow speed increases and reaches 0.7 the ratio τ/P decreases to about 2.7.This indicates that the plasma flow can be an essential parameter in determining the effectiveness of the damping mechanism as we concern the ratio τ/P.This effectiveness can be understood from Equations ( 16) and (23).As the flow speed increases, Equation (16) indicates that it causes the oscillation frequency to decrease, and Equation (23) indicates that it causes the damping rate to increase.These two variations together result in the strongly damped waves when the damping time is measured in units of the period time.Note that as the flow speed increases up to U = 0.7, our results in Figure 1 show that the ratio ω r /ω i never becomes smaller than 16 and the condition for using the perturbation method in this section, i.e., ω i = ω r , is satisfied.
It is interesting to note that the dispersion relation Equation (8) can be solved without assuming weak damping limit.To do this, Equation (8) can be solved to obtain four different wavenumbers k as functions of ω and the loop parameters from which two of them denoted by k 1 and k 2 are physically acceptable.Two of these solutions are acceptable because in the absence of damping mechanism they reduce to the wavenumbers given by Equation (12).These two physically acceptable wavenumbers can be subject to the footpoint condition k 2 − k 1 = 2nπ/L to give the complex oscillation frequency.Because the expressions obtained from Equation (8) for the wavenumbers are very awkward, we use one another way to obtain the exact solutions of the dispersion radiation.In this method, we extend the method used by Bahari (2017b)  These two equations can be solved numerically to obtain the complex values of k + and ω.This has been done here.
Interestingly the results obtained for τ/P are very close to those shown in Figure 1.The difference between the exact solutions and those obtained under the weak damping limit does not exceed one percent.Because of this, the diagrams of exact solutions are not shown in Figure 1.
One of the properties of the solutions determined in Equations ( 16) and ( 23) is that as we consider various background temperatures T 0 , the ratio τ/P for all the values of U scale with the same multiplier.To show this, we have plotted the normalized ratio of the damping time to the oscillation period defined by  16) and (23) that, because the normalized ratio defined here is the ratio of two values of the damping time to period ratio for the same temperatures, its dependence on the background temperature is removed and it does not depend on T 0 .It is also clear from the figure that for any background temperature as the flow speed increases from 0 to 0.7, the normalized ratio decreases from 1 to 0.2.In other words for any value of the background temperature, for this increase of the flow speed, the damping time in units of the period ratio decreases by a factor 1/5. Macnamara & Roberts (2010) has plotted the period ratio of fundamental mode to the first overtone of the longitudinal waves in static loops as a function of the dimensionless parameter   coronal loop L = 66 Mm, the loop temperature T = 1.6 MK the number density n = 10 9 cm −3 gives d = 0.015, which results P 1 /(2P 2 ) = 0.99.But for the hot SUMER loops with T = 6 − 10 MK, L = 100−200 Mm and n = 10 9 −10 10 cm −3 one finds d = 0.019−0.14,which is in a range where the period ratio well deviates from 1. Comparison of the solid and dashed curves in Figure 3 shows that introducing the plasma flow to the coronal loops causes the period ratio to decrease more than that of static loops and also to increase the range of d in which the period ratio well deviates from 1.

Conclusions
The oscillation and damping of the longitudinal waves in flowing coronal loops have been investigated.Thermal conduction is considered as the damping mechanism.In corona loop with constant plasma density and temperature, using the fact that the damping is weak, the equations of motion are solved using the perturbation method.In the leading order approximation, the oscillation frequency and the corresponding leading order eigenfunction of the standing longitudinal waves are determined.In the leading order approximation (for undamped waves), as proposed earlier by Bahari (2017b), Ruderman & Petrukhin (2019), and Bahari & Jahan (2020), the standing wave can be considered as the superposition of two propagating longitudinal waves.It was mentioned by Terradas et al. (2010) and Pandey et al. (2012) that in the absence of the directional symmetry the two propagating waves cannot have the same oscillation frequencies and the same damping rates.Based on this statement they concluded that in the absence of directional symmetry, the standing wave cannot be considered as the superposition of two propagating waves.In order to overcome this difficulty, we let both the oscillation frequency and the wavenumbers of the two propagating waves to be perturbed due to the presence of the damping mechanism, such that the footpoint boundary conditions are still satisfied.The result of this freedom in the wavenumber is that the two propagating waves can have the same oscillation frequencies and the same damping rates.Our results indicate that the ratio of the damping time to the oscillation period decreases by a factor of 1/5 as the flow speed increases from 0 to 0.7.Also the exact solution of the dispersion relation has been obtained without using weak damping assumption.Interestingly the exact solutions are approximately the same as those obtained using the perturbation method.Introducing plasma flow to the coronal loop causes the period ratio of the fundamental mode to the first overtone to decrease more.Also introducing the plasma flow causes the the range of the dimensionless damping parameter d in which the period ratio well deviates from 1 to increase.
the leading order amplitude of the waves.Imposing the boundary condition given by Equation (6) in z = 0 gives =

Figure 1 .
Figure1.The ratio of the damping time to the oscillation period of the longitudinal waves in a coronal loop as a function of the flow speed.Here κ 0 = 10 −11 and the background plasma temperature is considered T 0 = 10 6 K and T 0 = 1.5 × 10 6 K for the solid and dashed curves respectively.
to the case of obtaining the oscillation frequency of the damped waves.The dispersion relation Equation (8) is satisfied by ω and the wavenumber of the propagating wave that propagates in the positive z-direction k + we write as w dispersion relation is satisfied by ω and the wavenumber of the propagating wave that propagates in the negative z-direction k − can be written as wThe wavenumbers k + and k − are subject to the boundary condition k + − k − = 2nπ/L.This can be used to eliminate k − from Equation (25).Then we obtain value of T 0 in Figure 2. It is clear from Equations ( the damping mechanism can affect the period ratio.Here we have plotted the period ratio as a function of d for two different values of the flow speed in Figure3.The solid line is for the static loop with U = 0 and the dashed line is for the loops with U = 0.3c s .In this figure, the period ratio is determined by solving the dispersion relations given by Equation (26) numerically.For example, for a loop which its oscillations studied bySrivastava & Dwivedi (2011) for the length of

Figure 2 .
Figure2.The normalized ratio of the damping time to the oscillation period of the longitudinal waves in a coronal loop as a function of the flow speed.This diagram is relevant for all the loops with arbitrary values of T 0 and κ 0 .

Figure 3 .
Figure 3.The period ratio P P 2 1 2 as a function of d.The solid line is for the static tubes and the dashed curve corresponds to the loops with the flow speed U = 0.3.